Abstract

Balanced truncation of discrete linear time-invariant systems is an automatic method once an error tolerance is specified and yields an a priori error bound, which is why it is widely used in engineering for simulation and control. We present some new insight into this method. We derive a discrete version of Antoulas's $\mathcal{H}_2$-norm error formula \cite[p.218]{Ant05} and show how to adapt it to some special cases. This error bound is an a posteriori computable upper bound for the $\mathcal{H}_2$-norm of the error system defined as the system whose transfer function corresponds to the difference between the transfer function of the original system and the transfer function of the reduced system. The main advantage of our results is that we use the information already available in the balanced truncation algorithm in order to compute the $\mathcal{H}_2$-norm instead of computing one gramian of the corresponding error system. There is always a computational restriction on solving high-dimensional Stein equations for gramians. The a posteriori bound gives insight into the quality of the reduced system and can be used to solve many problems accompanying the order reduction operation.